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Mirrors > Home > MPE Home > Th. List > cmpfii | Structured version Visualization version GIF version |
Description: In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
cmpfii | ⊢ ((𝐽 ∈ Comp ∧ 𝑋 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑋)) → ∩ 𝑋 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6895 | . . . . 5 ⊢ (Clsd‘𝐽) ∈ V | |
2 | 1 | elpw2 5336 | . . . 4 ⊢ (𝑋 ∈ 𝒫 (Clsd‘𝐽) ↔ 𝑋 ⊆ (Clsd‘𝐽)) |
3 | 2 | biimpri 227 | . . 3 ⊢ (𝑋 ⊆ (Clsd‘𝐽) → 𝑋 ∈ 𝒫 (Clsd‘𝐽)) |
4 | cmptop 23223 | . . . . 5 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top) | |
5 | cmpfi 23236 | . . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅))) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝐽 ∈ Comp → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅))) |
7 | 6 | ibi 267 | . . 3 ⊢ (𝐽 ∈ Comp → ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)) |
8 | fveq2 6882 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (fi‘𝑥) = (fi‘𝑋)) | |
9 | 8 | eleq2d 2811 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (∅ ∈ (fi‘𝑥) ↔ ∅ ∈ (fi‘𝑋))) |
10 | 9 | notbid 318 | . . . . 5 ⊢ (𝑥 = 𝑋 → (¬ ∅ ∈ (fi‘𝑥) ↔ ¬ ∅ ∈ (fi‘𝑋))) |
11 | inteq 4944 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ∩ 𝑥 = ∩ 𝑋) | |
12 | 11 | neeq1d 2992 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∩ 𝑥 ≠ ∅ ↔ ∩ 𝑋 ≠ ∅)) |
13 | 10, 12 | imbi12d 344 | . . . 4 ⊢ (𝑥 = 𝑋 → ((¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅) ↔ (¬ ∅ ∈ (fi‘𝑋) → ∩ 𝑋 ≠ ∅))) |
14 | 13 | rspcva 3602 | . . 3 ⊢ ((𝑋 ∈ 𝒫 (Clsd‘𝐽) ∧ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)) → (¬ ∅ ∈ (fi‘𝑋) → ∩ 𝑋 ≠ ∅)) |
15 | 3, 7, 14 | syl2anr 596 | . 2 ⊢ ((𝐽 ∈ Comp ∧ 𝑋 ⊆ (Clsd‘𝐽)) → (¬ ∅ ∈ (fi‘𝑋) → ∩ 𝑋 ≠ ∅)) |
16 | 15 | 3impia 1114 | 1 ⊢ ((𝐽 ∈ Comp ∧ 𝑋 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑋)) → ∩ 𝑋 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ⊆ wss 3941 ∅c0 4315 𝒫 cpw 4595 ∩ cint 4941 ‘cfv 6534 ficfi 9402 Topctop 22719 Clsdccld 22844 Compccmp 23214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-om 7850 df-1o 8462 df-en 8937 df-fin 8940 df-fi 9403 df-top 22720 df-cld 22847 df-cmp 23215 |
This theorem is referenced by: fclscmpi 23857 cmpfiiin 41949 |
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